310 REPORTS ON THE STATE OF SCIENCE, ETC. 
These two equations require that 
Cj (Y) = FA(@?—7?)+Ba+Cy+D . : : rat (5) 
where A, B, C, D are constants. Thus 
dy 
Cy( — )=A: 
v(5) Awv+B 
by 
Cy(— )=—A C. 
? G y ) a 
6. Referring now to equations (7) and applying the operation Cy, we have 
2uCy(U) = —Y,+(1—o)[C— Ay] | (14) 
QuCy(V)= Xj+(l—a)[Ae+B]} * 
Thus Cy(U), Cy(V) are equivalent to a rigid body displacement consisting of a 
(small) translation of components ( — Y, + 1—oC)/2u parallel to # and (X, + (1 —c)B)/2u 
parallel to y and a small rotation (l—o)A/2y about the origin. 
This is Weingarten’s theorem (6). So soon as it is given that U and V have 
definite cyclic functions the result is obvious. For since U,, V, and U,V, are both 
solutions of the equations of elasticity, U,—U,, V,—V,, that is Cy(U) and Cy(V) are 
also solutions. But the stresses obtained from these must be zero, since the stresses 
are essentially acyclic. Thus Cy(U) and Cy(V), by a well-known result, are 
necessarily a rigid body displacement. 
We can now introduce Timpe and Volterra’s interpretation of Cy(U) and Cy(V). 
Reduce the multiplicity of connection by drawing a barrier CDE (fig. 11) from the 
Fie. 11. 
boundary of any inner hole H to the outermost boundary of the plate. 
This cut need not be straight. 
Now give the cut a very small rigid-body displacement so that it takes up the 
position C’D’E’. Cut the plate so that CDE is one boundary of the gap and C’D’E’ 
the other. If necessary, wedges of material may have to be cemented on to effect 
this. Now cement the two boundaries together, so that the plate once more forms a 
whole. Then, in consequence of this dislocation, strain is introduced, corresponding 
to displacements having for cyclic function with respect to the hole H the rigid body 
displacement which transformed CDE into C’D’E’. With regard to any other hole 
the displacements are still acyclic. 
In general there will be a cyclic function for each hole, each such function 
involving three constants, so that if there are m holes, so that the multiplicity of 
connection of the plate is n+1, these are 3n cyclic constants. By cutting 2 
channels of appropriate shape and recementing we obtain a plate under internal 
strain and stress, in which the displacements are affected by the correct cyclic 
functions. If now external forces are applied to the boundary of the cemented 
plate, these introduce a strain which involves only acyclic displacements, and 
so leaves the cyclic functions unaltered. 
We thus obtain a representation of the most general type of strain of such 
a plate as a combination of an external force system with a set of dislocations. 
BN 
